Combining Gompertzian growth and cell population dynamics

A two-compartment model of cancer cells population dynamics proposed by Gyllenberg and Webb includes transition rates between proliferating and quiescent cells as non-specified functions of the total population, N. We define the net inter-compartmental transition rate function: Phi(N). We assume that the total cell population follows the Gompertz growth model, as it is most often empirically found and derive Phi(N). The Gyllenberg-Webb transition functions are shown to be characteristically related through Phi(N). Effectively, this leads to a hybrid model for which we find the explicit analytical solutions for proliferating and quiescent cell populations, and the relations among model parameters. Several classes of solutions are examined. Our model predicts that the number of proliferating cells may increase along with the total number of cells, but the proliferating fraction appears to be a continuously decreasing function. The net transition rate of cells is shown to retain direction from the proliferating into the quiescent compartment. The death rate parameter for quiescent cell population is shown to be a factor in determining the proliferation level for a particular Gompertz growth curve.     

Calculation of Steel's cell loss factor and of the parameters of cellular kinetics for a population with an exponential growth of the cell number and cell death at G0 phase with a probability equal to 1

The method of calculation of three cell kinetics parameters (the Steel's cell loss factor phi, the proliferative pool Pc, and the mean number m of the proliferating cells after mitotic division of one cell) was shown to be the same for the exponential growth state of cell number with cell death at the G0-phase, and for the exponential growth state with cell death occurring immediately after mitosis. The value of the mean number delta of non-proliferating cells that appeared after mitotic division of one cell is different for these two models of the exponential growth state with the equal values of the other three parameters (phi, Pc, and m). A method is proposed for calculating the parameter delta on the data of the percentage of labeled cells obtained in the experiments with continuous cultivation of cells in the nutrient medium containing 3H-thymidine. The kinetics of cell line HL-60 (the experimental data of Foa et al., 1982) can be described at the first approximation, by a model of the exponential growth state with the cell death at the G0-phase, with Pc = 0.80, phi = 0.24, m = 1.61, delta = 0.39, and the life time of the non-proliferating cells tQ = 24 hours.     

Proliferation controls in a linear growth system: theoretical studies of cell division in the cartilage growth plate

The columnar arrangement of dividing cells in the epiphyseal cartilage plates of growing bones provides a model of a linear proliferation system. One factor which determines the rate of cell production, and hence the rate of growth, is the size of the proliferating population. In this one dimensional system this size is equal to the length of the proliferation zone. Two possible mechanisms for a differentiation control that sets a limit to the length of this zone have been tested in computer simulations. While a diffusion gradient control is consistent with cell kinetic measurements a division limit based on an inheritable growth substance is shown to require further development before the model fits experimental data.Cell division in the columns produces linear clones of cells. If the final length of a bone is set by a limit on the number of divisions that the cartilage stem cells can make, then the number of cells per clone is crucial in determining overall bone growth. The parameters that affect linear clone size have been investigated in computer simulations. Clone size depends largely on the relative division rate of stem cells to proliferation zone cells — but the data on stem cell division rates are generally unreliable.The analysis could be applied to other linear proliferating systems.     

Metabolism of proliferating and resting cells]

This review concerns the modern trends and experimental approaches to the study of cell cycle (cell population kinetics, cell structure and functions at various steps of the cycle,, etc.) and their input into the current views of cell proliferation controls. The resting state of the cell is considered and metabolic features of proliferating and resting cells are compared. Evidence is presented that resting cells are metabolically active and less resistant to the damaging factors that it has been previously supposed. The importance of this finding for biology and medicine, especially for cancer chemotherapy, is discussed.     

Using a stem cell and progeny model to illustrate the relationship between cell cycle times of in vivo human tumour cell tissue populations, in vitro primary cultures and the cell lines derived from them

There is increasing evidence that the growth of human tumours is driven by a small proportion of tumour stem cells with self-renewal properties. Multiplication of these cells leads to loss of self-renewal and after division for a finite number of times the cells undergo programmed cell death. Cell cycle times of human cancers have been measured in vivo and shown to vary in the range from two days to several weeks, depending on the individual. Cells cultured directly from tumours removed at surgery initially grow at a rate comparable to the in vivo rate but continued culture leads to the generation of cell lines that have shorter cycle times (1–3 days). It has been postulated that the more rapidly growing sub-population exhibits some of the properties of tumour stem cells and are the precursors of a slower growing sub-population that comprise the bulk of the tumour. We have previously developed a mathematical model to describe the behaviour of cell lines and we extend this model here to describe the behaviour of a system with two cell populations with different kinetic characteristics and a precursor–product relationship. The aim is to provide a framework for understanding the behaviour of cancer tissue that is sustained by a minor population of proliferating stem cells.     

Haemopoietic spleen colony growth: a versatile, parsimonious, predictive model

Abstract. Substantial support has been obtained for the stochastic model for stem cell differentiation first proposed by Till, McCulloch & Siminovitch (1964), over 20 years ago. By adding a cell maturation pathway, it is possible to predict (by computer simulation) the total number of cells and consequently the time at which individual colonies appear and disappear.
Only a few uncontroversial assumptions are required to predict that cells, uniform with respect to self-renewal, are capable of producing the high proportions of late disappearing and late appearing colonies observed experimentally in the spleens of irradiated mice that have been injected with normal haemopoietic cells. It is shown that differences in stem cell self-renewal only slightly influence the time of appearance of colonies; whereas changes in the kinetics of the maturing cells, by changing the size of colonies, has a marked effect on the time of appearance and disappearance of colonies and on the average doubling-time of colony-forming cells per colony (but not the doubling-time of individual colonies).
These results (1) seriously question the prevailing view that spleen colonies scored at 8 days measure a separate population (without the capacity for self-renewal), from those scored at 12 days; (2) argue against the existence of multiple sub-populations of stem cells with differing self-renewal and toxicity to cytotoxic agents; (3) help to identify those experiments for which it is obligatory to postulate heterogeneity, and (4) are consistent with self-renewal being regulated by a feedback control of stem cell differentiation, to which only proliferating stem cells can respond and where the stimulus for differentiation decreases at a time when the bone marrow is known to be depleted.     

Proliferation and differentiation of rat theca-interstitial cells: comparison of effects induced by platelet-derived growth factor and insulin-like growth factor-I

This study was designed to evaluate mechanisms regulating proliferation of steroidogenically active and steroidogenically inactive theca-interstitial (T-I) cells, and, specifically, to evaluate the effects of platelet-derived growth factor (PDGF) and insulin-like growth factor-I (IGF-I). T-I cells obtained from immature Sprague-Dawley rats were cultured in chemically defined media. Proliferation was assayed by thymidine incorporation and cell counting. Steroidogenically active cells were identified by the presence of 3beta-hydroxysteroid dehydrogenase activity. Flow cytometry facilitated separation of dividing cells (in S and G2/M phases of the cell cycle) from nondividing cells (in G0 and G1 phases of the cell cycle). PDGF alone (0.1-1 nM) produced a dose-dependent increase in DNA synthesis by up to 136%. IGF-I alone (10 nM) increased DNA synthesis by 56%. In the presence of both IGF-I (10 nM) and PDGF (0.1-1 nM), DNA synthesis increased by 108-214%. PDGF (1 nM) increased the total number of T-I cells by 43%; this effect was due to an increase in the number of steroidogenically inactive cells (47%). In contrast, the stimulatory effect of IGF-I (10 nM) was predominantly due to an increase in the number of steroidogenically active cells (163%). Separation of dividing cells from nondividing cells was accomplished with the aid of flow cytometry. In the absence of growth factors, the proportion of steroidogenically active cells was 35% lower among proliferating than resting cells. PDGF (1 nM) decreased the proportion of steroidogenically active cells among both proliferating and resting cells (by 43% and 16%, respectively). In contrast, IGF-I (10 nM) increased the proportion of steroidogenically active cells among proliferating cells by 56%. These findings indicate that differentiated/steroidogenically active cells divide; furthermore, PDGF and IGF-I may selectively stimulate proliferation of individual subpopulations of T-I cells, thereby providing a mechanism for development of structural and steroidogenically active components of the T-I compartment.     

Mathematical description and analysis of cell cycle kinetics and the application to Ehrlich ascites tumor

The linear and nonlinear aspects of the dynamics of the cell cycle kinetics of cell populations are studied. The dynamics are represented by difference equations. The characteristics of cell population systems are analyzed by applying the model to Ehrlich ascites tumor. The model applied for the simulations of the growth of Ehrlich ascites tumor cells incorporates processes of cell division, cell death, transition of cells to resting states and clearance of dead cells. Comparison of the results obtained with the model and the experimental data suggests that the duration of the mean generation time of the proliferating EAT cells increases with aging of the tumor. An attempt is made to relate the prolongation of cell mean generation time with processes of cell death and dead cell clearance. Studying the transition of cells to the resting states, it becomes apparent that in fact transition of proliferating cells to the resting states occurs somewhere close to the end of the cell cycle and with a rate that varies with the age of the tumor. Time course behavior of the cell age, cell size, and cell DNA distribution with aging of the tumor are obtained. Variations in average size and average DNA contents are determined.     

A comparative kinetic analysis of proliferation in vitro of Con-A-treated splenocytes and syngeneic leukaemia cells

Abstract. The growth kinetics of Con-A-treated mouse splenocytes and syngeneic leukaemia cells cultured in vitro were compared with respect to (i) the total cell number, (ii) the rate of [¹⁴C]thymidine incorporation (measured by pulse-labelling the cells at various times of incubation), and (iii) the labelling index of the cell populations. By correlating the thymidine incorporation, labelling index and cell number data, it has been established that, for both types of cells, the rate of [¹⁴C]thymidine incorporation is directly proportional to the number of cells synthesizing DNA. A new approach to cytokinetic analysis has been developed, showing that important information can be obtained by determining the cumulative kinetics of [¹⁴C]thymidine incorporation. The latter has been calculated by integrating the area underneath the time course of the rate of thymidine incorporation, and was directly proportional to the overall growth of both leukaemia cells and Con-A-stimulated splenocytes. Based on this proportionality, an estimate of the average duration of the S phase for both types of cells was calculated, suggesting that normal and neoplastic blasts maintain this parameter at a constant value (7.6 and 5.9 hr, respectively) throughout different stages of growth. The percentage of Con-A-responsive cells within the initial splenocyte population and their overall proliferation in vitro have been determined by a procedure which measures the cumulative kinetics of thymidine incorporation and the kinetics of cell total number in the presence or in the absence of the lectin, as well as in the presence of Con-A plus colcemid. A minor fraction (11%) of the initial splenocytes is recruited into cycle by Con-A, proliferating with similar kinetics to that of leukaemia cells in the same conditions. The great majority of the initial splenocyte population is unaffected by Con-A, decaying exponentially throughout the incubation with the same half-life (28 hr), both in the presence or in the absence of the lectin.     

A model of tumor growth based on cell cycle kinetics

This work describes a mathematical model of growth based on the kinetics of the cell cycle. A traditional model of the cell cycle has been used, with the addition of a resting (G₀) state from which cells could reenter the reproductive cycle. The model assumes that a growth regulatory substance regulates the transition of cells to and from the resting state. Other transitions between the phases of the cycle were modeled as a first order process. Cell loss is an important feature of growth kinetics, and has been represented by a general but tractable mathematical form. The resulting model forms a system of ordinary nonlinear differential equations. Analytic methods are employed first in the study of this system. Simplifying assumptions regarding cell loss give rise to special cases for which equilibrium solutions can be found. One special case, which assumes first order loss from all cell cycle phases at equal rates, is presented here. For small time values, approximations corresponding to exponential growth were developed. The equations describing an intrinsic growth rate were derived. Simulation methods were used to further characterize the behavior of this model. Parameter values were chosen based on animal tumor cell cycle kinetic data, resulting in a set of 45 model simulations. Several tumor treatment protocols were simulated which illustrated the importance of the intrinsic growth rate and cell loss concepts. Although the qualitative behavior regarding absolute and relative growth is reasonable, this model awaits data for model fitting, parameter estimation, or revision of the equations.     

RADIATION EFFECTS ON THE GROWTH RATE AND CELL POPULATION KINETICS OF ACTIVELY GROWING AND DORMANT ROOTS OF TRADESCANTIA PALUDOSA

Actively growing and dormant roots of Tradescantia paludosa were exposed to x-rays to compare the radiosensitivity of an actively proliferating tissue with that of one which is not active but is potentially proliferative. The level of effect was ascertained by the degree of change in the rate of root growth 4 days after exposure. Cell population kinetics were measured in control and in irradiated roots to determine whether or not a change was produced either in the number of proliferating cells or in the mitotic cycle duration which was sufficient to explain the altered rate of root growth. Nuclear volumes were also measured to provide an estimate of the relative total target size in actively growing vs. dormant roots. Tritiated thymidine was used to measure the cycle duration and the proportion of cells synthesizing DNA. The results showed that 184 and 305 r respectively were required to reduce the linear root growth rate to 37 per cent of that of the control for actively growing and dormant roots. Mitotic cycle duration, measured 4 days after x-ray exposure, was the same as in the control. The number of proliferating cells, however, was reduced. The rate of cell production in the irradiated roots was reduced to approximately one-half that of the controls. The average nuclear volumes of active and dormant roots were 733 and 491 µ³ respectively; thus the difference in the number of roentgens required to reduce growth to 37 per cent of that of the control can be attributed to the different average nuclear volumes. Therefore, the experiments suggest that part if not most of the differences in sensitivity between an actively dividing and an essentially non-dividing meristematic cell population resides in their different average nuclear volumes. Thus the law of Bergonie and Tribondeau needs to be reinterpreted, since the basic reason for the differences is secondary to whether or not the meristematic cells are proliferating.     

Control of macromolecular synthesis in proliferating and resting Syrian hamster cells in monolayer culture. 3. Electrophoretic patters of newly synthesized proteins in synchronized proliferating cells and resting cells

Electrophoretic patterns of newly synthesized proteins have been compared for hamster embryo fibroblasts in asynchronous cultures, mitotically synchronized cultures, and stationary phase cultures. Only proteins with molecular weight between 30,000 and 150,000, comprising 60–70% of the total cell proteins and excluding histones and collagen were included in the comparison. Although no significant differences could be detected between such patterns for cells at different stages of the cell cycle, significant differences were detected between patterns for cells in stationary phase and for proliferating asynchronous or synchronous cells in any stage of the cell cycle. These differences amounted to at least 5% of the newly synthesized cellular proteins. Much larger differences were detected between patterns from a nuclear fraction of proliferating and resting cells. These results indicate that normal cells in stationary phase are arrested in a state distinct from any phase of the normal cell cycle and may provide a biochemical marker for resting cells.     

Cell kinetics in a tumour cord

In some tumours, the viable cells grow around blood vessels forming cylindrical structures called tumour cords, which are surrounded by regions of necrosis. In the present paper, we propose a mathematical model for the cell kinetics in a tumour cord at the stationary state. Both proliferating cells and quiescent cells are considered, and the proliferating cell population is structured by age. Cell migration towards cord periphery is accounted for from a continuum viewpoint. The age distribution of proliferating cells, the fraction of cells in S phase, the growth fraction and the velocity along the cord radius are computed. The predictions of the model are compared with literature data obtained from two experimental rat hepatomas. The model was used to compute the profile of the oxygen tension within the cord. Possible modifications and extensions are also presented.     

A quantitative study of growth variability of tumour cell clones in vitro

Abstract.   Objectives : In this study, we quantify growth variability of tumour cell clones from a human leukaemia cell line. Materials and methods : We have used microplate spectrophotometry to measure growth kinetics of hundreds of individual cell clones from the Molt3 cell line. Growth rate of each clonal population has been estimated by fitting experimental data with the logistic equation. Results : Growth rates were observed to vary between different clones. Up to six clones with growth rates above or below mean growth rate of the parent population were further cloned and growth rates of their offspring were measured. Distribution of growth rates of the subclones did not significantly differ from that of the parent population, thus suggesting that growth variability has an epigenetic origin. To explain observed distributions of clonal growth rates, we have developed a probabilistic model, assuming that fluctuation in the number of mitochondria through successive cell cycles is the leading cause of growth variability. For fitting purposes, we have estimated experimentally by flow cytometry the average maximum number of mitochondria in Molt3 cells. The model fits nicely observed distributions in growth rates; however, cells in which mitochondria were rendered non-functional (ρ⁰ cells) showed only 30% reduction in clonal growth variability with respect to normal cells. Conclusions : A tumour cell population is a dynamic ensemble of clones with highly variable growth rates. At least part of this variability is due to fluctuations in the initial number of mitochondria in daughter cells.     

A model for persistent infection with Epstein-Barr virus: the stealth virus of human B cells.

Most adult humans are infected benignly and for life with the herpesvirus Epstein-Barr virus. EBV has been a focus of research because of its status as a candidate tumor virus for a number of lymphomas and carcinomas. In vitro EBV has the ability to establish a latent infection in proliferating B lymphoblasts. This is the only system available for studying human herpesvirus latency in culture and has been extremely useful for elucidating how EBV promotes cellular growth. However, to understand how EBV survives in the healthy host and what goes awry, leading to disease, it is essential to know how EBV establishes and maintains a persistent infection in vivo. Early studies on the mechanism of EBV persistence produced inconclusive and often contradictory results because the techniques available were crude and insensitive. Recent advances in PCR technology and the application of sophisticated cell fractionation techniques have now provided new insights into the behavior of the virus. Most dramatically it has been shown that EBV in vivo does not establish latency in a proliferating lymphoblast, but in a resting memory B cell. The contrasting behaviors of being able to establish a latent infection in proliferating B blasts and resting memory B cells can be resolved in terms of a model where EBV performs its complete life cycle in B lymphocytes. The virus achieves this not by disrupting normal B cell biology but by using it.     

Analysis of fibroblast growth factor influence on growth and developmental potential of rat foetuses in the in vitro culture model

The fibroblast growth factor's (FGF) influence on the growth and differentiation of 8- and 9- day-old rat foetus has been studied, whereas foetuses were grown in an in vitro culture model. Proliferation was analysed by the expression of proliferating cell nuclear antigen (PCNA). It was established that the usage of FGF in the first period of the culture lowers the growth no matter the foetus age at the moment of culturing and no matter whether it is a medium with or without a serum. If FGF is applied in the second culture period, it also lowers the growth, however younger foetuses in the in vitro culture model are more sensible to FGF negative influence. When FGF was applied in a lower concentration the growth of whole foetuses was improved in the in vitro culture model, which shows that the FGF influence on growth depends on the concentration. Stereological analyses have been done and showed that, in the in vitro culture model, FGF has no influence on proliferating cartilage tissue, but it stimulates the survival of nervel tissue cells. It has been shown that the quantitative research of growth processes in cultivated foetuses can precisely be done by combining classic methods of measuring whole foetus diameters and analysing the expression of proliferating antigen.     

Immunolocalization of proliferating perisinusoidal cells in rat liver

Summary There is now substantial evidence that perisinusoidal (Ito or fat-storing) cells are the principal source of extracellular matrix proteins during hepatic fibrogenesis. In rat liver these cells express the intermediate fiament protein desmin; this is now widely used as an immunohistochemical marker for these cells. It has been shown that in experimental models of acute and chronic liver injury there is an increase in the number of desmin-positive perisinusoidal cells prior to the deposition of matrix proteins; however, these studies have failed to establish whether local proliferation is involved in this expansion of the desmin-positive perisinusoidal cell population.In order to investigate the kinetics of the perisinusoidal cell response, we have developed a novel double-labelling immunohistochemical technique for the simultaneous demonstration of desmin and incorporated bromodeoxyuridine in proliferating perisinusoidal cells in sections of fixed paraffiin-embedded rat liver. Application of this technique to a model of acute liver injury (single dose carbon tetrachloride by gavage) has shown that expansion of the perisinusoidal cell population is contributed to by local proliferation, with a labelling index of 18.7% 2 days following injury.